DUODECIMALS. DUODECIMALS, or Cross Multiplication, is made use of by artificers in measuring their several works, and is performed by means of the following table : 1211' fourths make i third. 12"" thirds i second, 12" seconds 1 inch, 12' inches 1 foot. Glaziers, Masons, and others, measure by the square foot.-Painters, Paviors, Plasterers, &c., by the square yard.-Slating, tiling, flooring, &c., by the square of 100 feet.-Brickwork is measured by the rod of 162 feet, the square of wbich is 2721. See p. 42. RULE. (1) Arrange the terms of the multiplier under the same denominations of the multiplicand. (2) Multiply each term in the multiplicand, beginning at the lowest, * by the feet in the multiplier, aud write the result of cuch under its respective term, observing to carry one for every twelve. (3) Multiply, in the same manner, by the inches, and set the result of each term one place removed to the right-hand of those in the multiplicandot (4) Multiply then by the seconds, setting the result of each term two pluces removed to the right hand of those in the multiplicand. Multiply 9 ft. 4 in. 8 sec. by 5 ft. 8 in. 6 sec. I multiply by 5, saying 5 times 8 are 40, 4 and carry 3 ; 5 times 4 are 20 and 3 are 23, 11 and carry 1 ; 5 times nine are 45 and 1 are 46. · For the second 411 line I 8 times 8 are 64, 4 and carry 5, but the 4 over are thirds; and so of the rest. 7 4 9 8 8 46 11 3 1 say, 4 8 4 O'IH 53 NOTES + Feet * Hence the origin of the term cross multiplication, the operation being crossways, compared with multiplication in the common way. It is called Duodecimals, because the feet, inches, &c, are divided into twelve parts, whereas in decimals the unit is divided into tenths. multiplied into feet give feer. multiplied into inches give inches. Ex. 1. How much must I pay for a slab of marble 7 ft. 4 in. long, and 2 ft. i in. 6 sec. broad, at the rate of 7s. per square foot ? Ex. 2. What will be the expense of glass for a window that measures, in the clear, 10 ft. 61 in. in height, and 4 ft. 9 in. in width, at 1s. 9d. per foot ? Ex. 3. How much will a room cost in painting, at 9 d. per yard; the sides are 18 ft. 10 in. by 10 ft. 3 in., and the two ends are 16 ft. 6 in. by 10 ft. 3 in. ? Ex. 4. What shall I have to pay for statuary marble about my fireplace, at 14s. per foot; the hearth measures 6 ft. 4 in. by 2 ft. 3 in., the three fronts are each 4 ft. 2 in. hy 8 in., and the mantle-piece slab is 6 ft. by 9 in. ? Ex. 5. What will the paving of a court-yard come to, at 1s. 2d. per foot, the yard being 7.4 feet long, and 50 ft. 8 in. wide ? Ex. 6. How much shall I have to pay for slating a house, consisting of two sloping sides, each measuring 24 ft. 5 in. by 15 ft. 9 in. at the rate of 41s. per square of 100 feet? Ex. 7. What will the tiling of 10 houses come to, the roof of each house consisting of two sides, each 18 feet by 14, and ihe price of tiling at 285. per square ? Ex. 8. How many square rods are there in a brick wall 44 ft. 6 in. long, and 7 ft. 4 in. high, and 21 bricks thick ?* Ex. 9. If an oblong garden be 254 ft. 6 in. long, and 194 ft. 8 in. wide, what will a wall cost 10 ft. 6 in. high, and 2 bricks thick, at 151. 15s. per square rod ? Ex. 10. How much shall I have to pay for the plate-glass of four windows ; each window consists of 16 panes, and each pane measures 204 inches by 15 inches at 9s. Od. per foot? Ex. 11. How many solid feet of fir are there in a piece of timber 35 ft. 4 in. long, and 13 inches by 14 inches ? + Ex. 12. How many solid feet of oak are there in a piece 14 feet 3 inches long, and 2 feet 101 by 2 feet 2 inches ? Ex. 13. How many solid feet of fir are there in 46 joists, each 14 feet 3inches long, 7 & inches deep, by 3 inches broad? NOTES. * Bricklayers value their work at the rate of a brick and a half, or tliree half bricks thick; and if the wall be more or less than this, it must be reduced to that thickness by the following rule :-? Multiply the measure found by the number of half bricks, and divide by three :" ihus, if the wall be 2 bricks thick, I multiply by 5, and divide the product by 3. Ex. If the wall be 50 feet long, and 9 high, and 2 bricks thick, it 600 will be 50 X 9 X = 600 feet; and : 21 squ. rods nearly. 272 + Carpenters' rules are divided into eighths, so that in these cases the eighths must be reduced to twelfths, or the whole must be worked by decimals. In this and the following questions, the length, breadth, and thickness, must be multiplied into one another. 4 3 GEOGRAPHICAL CALCULATIONS. 3 feet make 1 yard, 1760 yards 1 mile, 694 miles, or 60 geographical miles i degree.* The degree is usually reckoned in round numbers, at 691 miles; but if accuracy be attended to, the nuniber in the table is too large : the real length of a degree is 365194 English feet, or 69 miles 288 yards: this has been ascertained by actual measurement, so that the circumference of the earth is equal to 69 miles, 288 yards X 300 (because in every circle there are 360 degrees) = 25000 miles nearly. Geographers reckon on the globe two kinds of degrees, viz. degrees of latitude, and degrees of longitude.* The degrees of latitude, which are measured, from north to south, on the meridian, are all of one length, as above. But the degrees of longitude, or the circles which pass round the earth in each parallel of latitude, continually diminish in proceeding from the equator towards the Poles, but at the equator they are of the same length as those of latitude. The following is a Table of the length of the degrees of longitude, carried to three places of decimals, in every 5 degrees of latitude. TABLE. Lat. Eng. miles. Lat. Eng. miles. Lat. Eng. miles. Lar. Eng, m. Here it is evident, that at latitude 40°, the degree is little more than 53 miles in length; at 70° it is only 23 miles; and at the pole, or goo, it comes to nothing, it being supposed to be a point. 1. To find the distance, in iniles, between airy two places, having the same degree of latitude. Rule. Having found the distance between the places, in degrees, multiply the number so found by the number in the table opposite the given degree of latitude.. Ex. How many miles distant is Madrid, in Spain, from Bursa, in Natolia; the latitude of both is 40° N., but the long. of Madrid is about 30 W., and that of Bursa 29° E. ? NOTE. * Longitude expresses the distance of meridians, or circles, which are supposed to pass over the head from north to south; and latitude expresses the distance of a place north and south from the equator. The difference in longitude is 3° + 29° = 32', this multiplied by 53.01, the number of miles in a degree at the given latitudes, gives 1696 for the miles beļween Madrid and Bursa. II. To find the distance between any two places, having the same degree of longitude. Rule. Multiply the number of degrees between the places by 69.2, and the answer is in miles. Ex. How far is London from Mount Atias in Africa, the former is 511° N. L., the latter 311° N. L.? Here the distance is 20°, and 20 X 69.2 -- 1384 miles. TIME is measured by the revolution of the earth about its axis : every revolution is completed in 24 hours; and as 3600 15° of motion answers to 60' in time, or 1 hour. 4' J. To convert time into motion. Rule. Multiply the hours by 15, and divide the minutes by 4, and the answer is in degrees, &c. Thus 4 h. 20 min. in time, answer to 65° in motion. 11. To convert motion into time. Rule. Divide the given number of degrees of motion by 15, and the answer is in time. Thus 65° of motion answer is 4b. 20 inin, in time : 15)65(4 24 60 5 15)300(20 300 Ex. 1. What o'clock is it at Athens, which is 23° 57' east longitude of London, now it is 12 at the metropolis ? Athens being east of London, the clocks there will be before the clocks here. 15)23° 57'(1 hour. Answer. When it is 12 o'clock at 15 8 60 15) 537(36 min. nearly.. Ex. 2. What o'clock is it at Philadelphia in America, now it is 12 at London ? Philadelphia is 75° 8' west longitude of London, of course the clocks there are behind those here. 15)75° 8'(5 bo. o min. 32 sec. In this case the answer is. 12 h.5 h. om. 32 s. = 6h. 59 m. 28 s. 60 or very nearly 7 in the morning. 15) 480(32 In many maps the longitude is reckoned from Ferro, ono of the Canary Islands, which is 17° 45' west of London. III. To reduce the longitude of Ferro to that of London. Rule 1. If the place be east of London, subtract from it 17° 30', and the remainder is the longitude east of London. Thus, from Ferro, Constantinople is 46° 44'; to reduce this to the longitude reckoned from the meridian of London, we say, 46° 44' — 17° 45' = 28° 59'. 2. If the place be west from Ferro, add to the given longitude 17° 45'. Thus, Boston is 52° 48' west of Ferro, but it is west of London 32° 48' + 17° 45' = 70° 33'. 3. If the place lies between Ferro and London, its longitude will be obtained by subtracting its longitude east of Ferro from 17° 45'. Thus, Lisbon is go 40' east of Ferro, and it is west of London 17° 45' . 8° 40' = 9° 5'. By a reverse method may be reduced the longitude from London to that of Ferro. The earth being globular, it is a useful problem to ascertain the extent of the visible horizon : or IV. To find the distance to which a person can see at any given height of the eye. RULE. Multiply the square-root of the height of the eye, in feet, by 1.2247, and the product i; the distance in miles to which we can see from that height. See p. 143. Ex. 1. How far can a sailor see, standing at the topmast of a ship, 144 feet high? The square-root of 144 is 12 ; therefore 1.2247 x 12 = 14.7 miles. Thus, in this situation, a sailor might, on a very clear day, descry land at the distance of 5 leagues, nearly ; and he might see the top-mast of another ship at a still greater distance. Ex. 2. To what distance could a person see from the top of St. Paul's, which is 340 feet high ? 340 X 1.2247 = 18.44 X 1.2247 = 22.58 miles, or something more than 22 miles. |